James A. Pennline

Glenn Research Center, Cleveland, Ohio

Lealem Mulugeta

Universities Space Research Association

Division of Space Life Sciences, Houston, Texas

Evaluating Daily Load Stimulus Formulas in

Relating Bone Response to Exercise

NASA/TM—2014-218306

June 2014
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James A. Pennline

Glenn Research Center, Cleveland, Ohio

Lealem Mulugeta

Universities Space Research Association

Division of Space Life Sciences, Houston, Texas

Evaluating Daily Load Stimulus Formulas in

Relating Bone Response to Exercise

NASA/TM—2014-218306

June 2014

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
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Level of Review: This material has been technically reviewed by technical management. 

This report contains preliminary fi ndings, 

subject to revision as analysis proceeds.
NASA/TM—2014-218306 1 

Evaluating Daily Load Stimulus Formulas in 

Relating Bone Response to Exercise 

James A. Pennline 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Lealem Mulugeta 

Universities Space Research Association 

Division of Space Life Sciences 

Houston, Texas 77058 

Abstract 

Six formulas representing what is commonly referred to as “daily load stimulus” are identified, 

compared and tested in their ability to relate skeletal mechanical loading to bone maintenance and 

osteogenic response. Particular emphasis is placed on exercise- induced skeletal loading and whether or 

not the formulas can adequately capture the known experimental observations of saturation of continuous 

cyclic loading, rest insertion between repetitions (cycles), recovery of osteogenic potential following 

saturation, and multiple shorter bouts versus a single long bout of exercise. To evaluate the ability of the 

formulas to capture these characteristics, a set of exercise scenarios with type of exercise bout, specific 

duration, number of repetitions, and rest insertion between repetitions is defined. The daily load values 

obtained from the formulas for the loading conditions of the set of scenarios is illustrated. Not all of the 

formulas form estimates of daily load in units of stress or in terms of strain at a skeletal site due to the 

loading force from a specific exercise prescription. The comparative results show that none of the 

formulas are able to capture all of the experimentally observed characteristics of cyclic loading. However, 

the enhanced formula presented by Genc et al. does capture several characteristics of cyclic loading that 

the others do not, namely recovery of osteogenic potential and saturation. This could be a basis for further 

development of mathematical formulas that more adequately approximates the amount of daily stress at a 

skeletal site that contributes to bone adaptation. 

Introduction 

Quantifying the adaptive response of bones to loading in terms of a mechanical stimulus magnitude 

related to stress/strain from weight bearing activities is a modeling activity originated from studies by 

Carter et al. (1987) and Whalen et al. (1988). Commonly referred to as a Daily Load Stimulus (DLS), a 

stimulus expression was used in a computational simulation of a theory to determine the distribution of 

bone density in the adult proximal femur (Beaupre et al., 1990), and expressions were fit to experimental 

data in animal studies (Turner, 1998). Various versions have been developed and used for different 

research applications, such as the use of a stimulus magnitude formula to relate to the rate of fatigue 

damage accumulation and its influence on bone remodeling activation (Hazelwood et al. 2001, Nyman 

et al. 2004, and Hazelwood and Castillo 2007). Other examples include the use of a stimulus formula for 

estimating and comparing the osteogenic response of bone to different amounts of daily activity (Genc 

et al., 2009), as well as to estimate the effectiveness of daily exercise induced loading as countermeasure 

to spaceflight bone loss (Cavanagh et al., 2010). The latter application is used as a primary motivation for 

this perspective. 

 
NASA/TM—2014-218306 2 

Under the conditions of microgravity, astronauts lose bone mass at a rate of 1 to 2 percent a month, 

particularly in the lower extremities such as the proximal femur (Lang et al., 2004, Buckley, 2006; 

LeBlanc et al., 2007). The most commonly used countermeasure against bone loss in microgravity has 

been a prescribed exercise regime (Layne and Forth, 2008). However, data has shown that existing 

measures of exercise do not completely eliminate bone loss in long duration, 4 to 6 months, spaceflight 

(Lang et al., 2004; LeBlanc et al., 2007; Keyak et al., 2009; LeBlanc et al., 2013). The introduction of the 

Advanced Resistive Exercise Device (ARED), coupled with improved nutrition, has further reduced the 

4 to 6 month bone loss (Smith et al., 2012). But further work is needed to implement optimal exercise 

prescriptions since bone loss has not been completely eliminated, and future exploration class missions 

can only accommodate exercise devices that will be much smaller than the ARED. In this light, NASA’s 

Digital Astronaut Project (DAP) is working with physiologists in the NASA Human Research Program to 

implement well-validated computational models to help predict and assess bone loss during spaceflight, 

and enhance exercise countermeasure development. More specifically, computational modeling is 

proposed as a way to augment bone research and exercise countermeasure development to specifically 

target weight-bearing skeletal sites that are most susceptible to bone loss in microgravity. 

Several bone remodeling theories, a research area that has seen increased activity in the last decade, 

have been developed in an attempt to quantify the contribution of mechanical stimulus from daily loading 

to bone homeostasis via mechanotransduction (Hernandez et al., 2000, Garcia-Aznar et al., 2005). There 

have also been models that attempt to connect mechanical stimulus to cellular dynamics (Maldonado 

et al., 2007; Pennline, 2009). The major unit of the temporal scale in these modeling theories is typically a 

24 hour day and the biological rate constants tend to be in units of ‘per day’. For the model under 

development in the DAP, we are interested in establishing an expression that can approximate the amount 

of daily stress/strain at specific skeletal sites from exercise prescriptions, and in understanding how those 

stress/strain levels translate to maintenance of bone via bone remodeling, particularly in a weightless 

environment, in order to design optimal exercise prescriptions. 

The purpose of this article is to identify the various DLS expressions that have been developed to 

date, conduct a perspective comparison, and test their abilities as appropriate tools to quantify stress/strain 

from exercise induced loading at specific skeletal sites. We describe a simple set of exercise scenarios and 

examine how the formulas carry out their evaluations in terms of the exercise performance logistics. 

Specific attention is given to their ability to capture the experimentally observed effects of cyclic loading. 

These include, for example, the diminishing effect of continuous repetitions, increased effect of rest 

insertion between repetitions of an exercise, and increased effect of rest insertion between entire bouts of 

an exercise. 

Formulas and Explanation of Their Differences 

Key experiments in the past have given insight into the dynamic nature of skeletal loading and helped 

to suggest the formulation of DLS expressions. The experimental study by Rubin and Lanyon (1985) on 

the isolated ulna of roosters demonstrated that newly formed bone area is proportional to applied strain 

while the same type of experiment (Rubin and Lanyon 1984) showed that increasing loading duration past 

36 cycles per day caused little further increase in bone mass. Turner et al. (1994) showed that increasing 

the applied loading frequency of four-point bending significantly increased the mechanically induced 

bone formation rate on the endocortical surface of the rat tibia. In a similar experiment (Turner et al. 

1995), mechanically induced bone formation rate was proportional to peak to peak loading magnitude at 

2 Hz. Evidence of the diminishing effect of increasing loading duration was seen in an experiment by 

Umemura et al. (1997) which related increased bone mass to the number of jumps per day that rats were 

trained to perform. 

In an article that cites these experiments, Turner (1998) translated the observations into rules for bone 

adaptation to mechanical stimuli. One rule states that bone adapts to dynamic skeletal loading, i.e., strain 

stimulus increases with increasing magnitude or frequency of the dynamic loading, and strain stimulus is 

 
NASA/TM—2014-218306 3 

enhanced by increasing strain rate. Another rule is that continuous loading duration has a diminishing 

effect on bone formation response. Based on these rules, he discussed mathematical formulas that can be 

related to a DLS. One expression, which he credits to Carter et al. (1987), that emphasizes the effect of 

load magnitude and load frequency is given as: 

m k

j

m S N j j

1/

1 















	 
 



 (1)

The quantities that the symbols represent are: 

S Stimulus 

k Number of loading conditions/activities (can be thought of as number of exercises per day) 

N j Number of loading cycles per loading condition (or number of repetitions per exercise) 

j Effective stress (or strain) per loading condition 

m Weighting factor (experimentally determined) 

Beaupre et al. (1990) assumed a value of m = 4 based on correlation with available experimental data by 

Whalen et al. (1988) and a study of experimental data of Rubin and Lanyon (1984), while Turner 

estimated a value of 3.5 based on the data of Umemura et al. (1997). Note that the expression represents a 

quantity that is proportional to what could be interpreted as a stimulus to bone that affected bone 

formation rate or an increase in bone mass. 

The work of Carter et al. (1987) involved more specific relations to bone properties than what is 

indicated by Equation (1). They defined a quantity * S as the constant daily stimulus that keeps a region 

of bone from experiencing neither a net loss nor gain of bone apparent density. Assuming a relation to a 

linear superposition of stimuli for all loading activities in a day it was expressed as: 

( / ) per day 

1

* m j ult

k

j

N j S 	 
   



(2)

where ult is the ultimate effective stress. Subsequently, assuming that the bone maintenance stimulus is 

constant everywhere, the ultimate stress is factored out of the summation and solved for ult to obtain the 

following relation, 

m k

j

m ult N j j

1/

1



















 	 
 



(3)

Referring to the work of Carter and Hayes (1977), which showed the bone strength ult for a single 

loading to failure to be approximately proportional to the square of the apparent density  (i.e., 

1/ 2 ( )  	 ult ), Carter et al. assumed the following approximation for the apparent density for multiple 

loading conditions, 

m k

j

m N j j

1/ 2

1



















 	 
 



(4)
NASA/TM—2014-218306 4 

In addition to the daily loading history based on stress and bone strength, Carter et al. also formulated one 

based on fatigue damage accumulation, and another based on bone strain energy density. 

Genc et al. (2009) used expression (4) to begin their development of an activity determination 

algorithm based on entire days of in-shoe-forces such as those resulting from typical weight bearing 

locomotion activities like walking, running, and standing. 

2m

1

per day 1

DLS ( ) 

















 




k

j

m N j Gzj (5)

Here, the effective stresses, j , have been replaced with peak ground reaction force (GRF) magnitudes, 

Gz j , in the late stance phase for each loading cycle. The term in-shoe force refers to data collected by an 

ambulatory biomechanical data collection system for use in space (Cavanagh et al., 2009). The goal of the 

whole effort was to have a daily load model that could be used in measuring the daily load impacted to 

the musculoskeletal system of an astronaut during a typical day on earth (Cavanagh et al., 2010). The 

earth DLS would serve as a target for prescribing a similar dose through exercises during spaceflight. 

Note that the exponent is different from the exponent in the expression of Equations (1) and (3). From the 

outline of the stress analysis (Eqs. (2) to (4)), it is clear that the DLS expression of Genc et al. is directly 

related to bone apparent density, whereas Turner’s DLS expression relates to bone strength. 

From Equation (5), an enhanced DLS (EDLS) was developed which, when applied to an entire day of 

loading activities, takes into account, 

 Saturation of osteogenic response of bone cells to continuous cyclic loading, 

 Mechanosensitivity recover time between repeated cyclic bouts, and 

 Contribution of low microstrain loads (e.g., standing postural shifts), 

2m

1

per day 1

EDLS ( ) 

















 
 



N Gz AL

k

j

m j j (6a)

AL  (23.0)(Standing time) (6b)

The additional term AL represents accumulated load from standing which was derived from an 

assumption that AL from 1 min of standing equals 25 percent of the AL from 1 min of slow walking. 

The idea of replacing the effective stresses by GRF was credited to a study by Whalen et al. (1988) in 

which the resulting DLS expression was used to study the effects of locomotion activities on calcaneal 

bone maintenance. However, that study did not take into account the effects of saturation, recovery time, 

and low microstrain. 

Note that saturation and recovery parameters do not explicitly appear in the formula, Equation (6). 

Instead, a 5 min saturation limit for running and a 20 min limit for walking were identified in 

experimental results from animal studies (Robling et al., 2002). Also, a recovery factor based on a fit to 

animal data (Robling et al. 2001) was used and is given by: 

1  t / 2

e  (7)

where t is the time in hours since the end of the previously saturated period. Then, an algorithmic flow 

chart for calculating the EDLS was outlined. It included decision branches that tested whether or not each 
NASA/TM—2014-218306 5 

cycle from a continuous load exceeded the limits. The 5 and 20 min limits for continuous running and 

walking in a single bout are implemented by excluding any GRF in the summation part of the formula 

that occurs after the limits. To implement the recovery, subsequent ground reaction peaks during periods 

of running or walking after saturation are multiplied by their recovery factor. 

Referring again to the work of Turner (1998), more variations in the DLS expression are formulated. 

He observed that the diminishing effect of continuous loading duration from the studies of Rubin and 

Lanyon (1984) and Umemura et al. (1997) could be mathematically modeled with a logarithmic function. 

Combining these observations with observations that both magnitude and strain rate can play a role in 

determining bone adaptation (Turner et al., 1995; Rubin and McLeod, 1994), the following alternate DLS 

expression is suggested: 






	 

k

j

N j E j S

1

log(1 ) (8)

The quantities k and N j have the same definition as Equation (1). The new quantity  is intended to 

take into account a strain rate contribution to the DLS, and is defined as 

ji

n

i

j ji E 
 f



 

1

 (9)

where the variables are defined as follows: 

n Number of frequency components for each loading cycle within a loading condition 

ji f Frequency component i for each cycle waveform in loading condition or exercise, j

ji  Peak to peak strain for strain component waveform i for each cycle N j . 

An application of formula (8) was used to quantify the load from a single bout of continuous bilateral 

jumping performed by human subjects (Ratalainen et al., 2009). 

In addition to Equation (8), Turner (1998) replaced the stress factor in Equation (1) by the strain 

factor of Equation (9) obtaining another DLS version: 

m k

j

m S N jEj

1/

1 















	 




 (10) 

Turner argued that, in the case of a long bone subjected to a periodic cyclic load represented by a 

single sinusoidal wave term with a frequency f, the strain rate is proportional to f where  is the peak to 

peak dynamic strain, which is proportional to twice the amplitude of the sinusoidal wave. Note that this 

holds for a cyclic load represented by a single sinusoidal wave. The suggestion was made that the strain 

from a cyclic load represented by a general loading wave form, e.g., an exercise, could be expanded in a 

Fourier series consisting of sinusoidal waves at different amplitudes and frequencies. The derivative of 

the Fourier series, strain rate, would eliminate the static term and leave an infinite series of sinusoidal 

terms. This lead to the assumption that the strain stimulus could be proportionally represented as 

i i i E 
 f 

 	 

1 (11) 

where i is the strain magnitude related to the amplitude of term in the expansion and fi is the frequency 

of term i. Since the daily load consists of k activities or loading conditions, Equation (9) becomes the 
NASA/TM—2014-218306 6 

strain stimulus as a factor in the overall mechanical stimulus expression of Equation (8) or (10). 

Unfortunately, it turns out that an infinite series formed as in Equation (11) may diverge. In such cases the 

series cannot possibly represent the strain stimulus in a proportional sense. 

Mathematical models of bone adaptation seen in the first decade of this century have also used 

versions of a daily load. Because of the similarity to Equations (1) and (3), the versions used by 

Hazelwood et al. (2001) and Hazelwood and Castillo (2007) are mentioned here. Referencing Carter et al. 

(1987), Hazelwood et al. (2001) used the formula given by: 

Li

n

i

m 
si R



 

1

 (12) 

where the following definitions apply: 

n Number of loading conditions (or exercises), 

is Strain range for each loading condition i, 

Li R Cycles per day for each loading condition i 

This expression appears to be more closely related to the daily loading history based on fatigue 

damage accumulation (Carter et al., 1987) where cyclic stress range is used explicitly in place of strain 

range. Equation (12) was treated as a loading potential proportional to fatigue micro damage rate in a 

study of bone remodeling due to overload. Similar to previous studies already mentioned, the value of m

was chosen as 4 but other values (1, 6, and 8) were also examined. 

For simplicity, cyclic loading of only one magnitude was applied (n = 1) and s was assumed to 

represent the principle compressive strain which returns to zero at the end of each cycle. Hazelwood and 

Castillo (2007) also used Equation (12) with m = 4 as the local stimulus for bone remodeling in a 

mechanistic model integrated with a finite element model of the femur. Specifically, they studied the 

effects of loading characteristic of marathon training on bone density, remodeling, and microdamage of 

the femur. In this case s was assumed to be the principle strain component with the largest magnitude. 

The DLS expressions discussed above are summarized in Table 1. 

TABLE 1.—SUMMARY OF DLS EXPRESSIONS 

DLS Expression Reference 

I 

m k

j

m N j j

1/

1 
















 



Carter et al. (1987) 

II .

2m

1

per day 1 


















 



k

j

m N j j

Carter et al. (1987) 

Whalen et al. (1988) 

III ( )

2m

1

per day 1 


















 



N Gz AL

k

j

m j j

Genc et al. (2009) 

Cavanagh et al. (2010) 

IV 




k

j

N j E j

1

log( 1 ) Turner et al. (1998) 

V 

m k

j

m N jEj

1/

1






















Turner et al. (1998) 

VI Li

n

i

m


si R

1

Hazelwood et al. (2001) 

Hazelwood & Castillo (2007) 
NASA/TM—2014-218306 7 

In addition to the works mentioned above, a number of experimental studies have revealed additional 

insight into the nature of dynamic loading. Umemura et al. (2002) conducted a study with rats trained to 

perform a jumping exercise in three different formats: 

(a) a single bout of 20 continuous jumps separated by 3 sec interval between jumps, 

(b) a single bout of 20 jumps separated by 30 sec interval between jumps, and 

(c) 2 bouts of 10 continuous jumps separated by 6 hr. 

The results of this study revealed that the rats in all three groups had greater bone masses per body weight 

of the femur and tibia than rats in the control group which were allowed normal cage activity. However, 

those values in groups (b) and (c), although similar to one another, were greater than in group (a). A more 

recent experiment in which mice were subjected to cyclic loading using a noninvasive murine tibia 

loading device revealed that periosteal bone formation rate was significantly increased with 10 sec rest at 

zero load between each load cycle compared with continuous cyclic loading (Srinivasan et al., 2007). 

Results of these experiments suggest the assumption that rest inserted loading at more than a few seconds 

between cycles in a single bout of loading is more beneficial than continuous cyclic loading. Also, 

multiple shorter bouts of an exercise in a day, separated by sufficient time to reinitiate osteogenic 

sensitivity, are more beneficial than one long bout. 

Results of Application to Exercise 

Based on the results of the experimental studies mentioned above, the following assumptions 

regarding cyclic loading are suggested: 

A. Bone adaptation and maintenance are functions of dynamic loading (Turner et al., 1998). 

B. Osteogenic potential decreases with increasing repetitions and eventually saturates (Rubin and 

Lanyon, 1984; Robling et al., 2002). 

C. Once osteogenic potential has significantly diminished or saturated, a minimum time is required 

for recovery of full cellular sensitivity (Robling et al., 2001). 

D. Multiple shorter bouts of an exercise are more beneficial than one long bout, if there is sufficient 

time for recovery of cell sensitivity between bouts (Umemura et al., 2002). 

E. Within a weight bearing exercise, rest insertion of possibly 10 to 30 sec between repetitions is 

more beneficial than continuous repetitions (Srinivasan et al., 2007). 

To test how well each DLS expression captures the characteristics of cyclic loading listed in the 

assumptions, we consider the following test set of exercise schemes and examine how the DLS 

expressions compute. 

(1) Running on a treadmill for 18 min (Fig. 1.)

(2) A single bout of 20 squats with 3 sec pause between repetitions (Fig. 2) 

(3) Two bouts of 10 squats separated by 6 hr., with 3 sec pause between repetitions 

(4) A single bout of 20 squats with 10 sec rest insertion between squats 

(5) One bout of 10 squats followed by another bout of 10 squats 1 hr later 

Each scheme is assumed to be the only activity performed in a single day. The same subject is assumed to 

perform the exercises, thus keeping the influence of the anthropometrics consistent. For each single bout of 

squats, the individual is assumed to be under the maximum possible load that avoids muscle failure. The 

skeletal site of interest can be assumed to be the calcaneus of one foot for exercise scenario (1) while the site 

of interest can be taken as the proximal femur or just the femoral neck for exercise scenarios (2) to (5). 
NASA/TM—2014-218306 8 

TABLE 2.—EVALUATIONS OF THE EXERCISE SCENARIOS FOR EACH OF THE DLS EXPRESSIONS 

DLS Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 

I   m m

r

1/

1200   m m

s

1 /

20   m m

s

1 /

20   m m

s

1 /

20   m m

s

1 /

20 

II  1/(2 ) 1200 m m r   1/(2 ) 20 m m s   1/(2 ) 20 m m s   1/(2 ) 20 m m s   1 /(2 ) 20 m m s 

III   m m Gz r 2

1

333( )   m m Gz s 2

1

20 ( )   m m Gz s 2

1

19.5( )   m m Gz s 2

1

20 ( )   m m Gz s 2

1

14( )

IV (1) log(1201) E (2) log(21)E (3) log(121) E (4) log(21)E (3) log(121)E

V   m m E 1/

(1) 1200   m m E 1/(2) 20   m m E 1/(3) 20   m m E 1/(4) 20   m m E 1/(5) 20

VI 1200 m

rs  20 m

s s 2  10 ms s  20 ms s 2  10 ms s

Table 2 shows how each DLS expression would form its numerical value based on the logistics of 

each exercise scheme. Each value in the table has the unit of per day as defined by the DLS expressions. 

The following notation can be matched to the variables in the summary of the expressions: 

r Effective stress on bone in the calcaneus from GRF of each impact during running 

s Effective stress on bone in the proximal femur from each squat 

Gzr Peak GRF during running 

Gzs Peak GRF from the squat exercise 

rs Maximum principle strain in the calcaneus during running 

ss Maximum principle strain in the proximal femur from a squat 

Column 1 shows the DLS evaluations for the running exercise. Referring to Figure 1, the stride period 

is about 0.9 sec. In 18 min, the number of impacts (cycles) per foot amounts to 1200 cycles. For DLS III, 

the implemented algorithm eliminates the contribution from any impact occurring after 5 min which 

limits the number of cycles from this exercise bout to about 333. For DLS IV and V, the symbol E(1)

represents the sum of a finite number of products of amplitude and frequency, Equation (11), from the 

Fourier expansion of one cycle which, for the left foot, is illustrated by the cycle beginning at about 0.14 

and ending at 1.04 sec. 

Column 2 shows the DLS evaluations for a single bout of 20 squats performed continuously. In this 

case, the symbol E(2) in IV and V represents the sum of a finite number of terms, Equation (11), formed 

from the products of amplitude and frequency in the Fourier expansion of a strain waveform of one squat. 

Figure 2 shows the magnitude of the resultant force at the hip joint for one repetition or one squat 

assuming a load on the individual that results in a joint force at the hip equal to about body weight prior to 

beginning the squat. It was produced relative to the data from Bergmann et al. (2001) showing the joint 

contact force at the hip from an individual performing a knee bend while presumably unloaded. A finite 

element model of the proximal femur could use the force vector to compute effective stress and maximum 

principle strain. 

The single activity in scenarios 1 and 2 reduce the expressions in Table 1 to a single term, k = 1. In 

the case of scenario 3, k = 2, and the bouts, although identical, are counted as two individual activities 

which is reflected in the evaluations in column 3 of Table 2. Note that DLS III multiplies the second bout 

of 10 squats by the osteogenic recovery factor (1–exp (–3)), Equation (7). Expression IV shows the 

additive properties of the log function when combining two separate bouts of 10 squats. Note also that if 

squats are performed identically, E(3)  E(2) . 
NASA/TM—2014-218306 9 

The forth squat exercise, although performed with a 10 sec rest insertion between repetitions, is 

treated as a single activity (k = 1 in the expressions). The results turn out to be identical to the evaluations 

for scenario 2 (column 2). However, the summation E(4) may be different from E(2) . 

Since the second exercise bout in scenario 5 is performed 1 hr after the first bout, the DLS 

expressions treat this as two loading conditions (k = 2). The contribution to the DLS from all loading 

cycles in DLS III is only 14. The reason for the difference is because the cycles from the second bout of 

squats are multiplied by the osteogenic recovery factor (1  exp( 0.5)) . The summation E(5) should be 

the same as E(2) since the squats are performed identically with no rest insertion. 

Figure 1.—This figure was plotted using, with permission, example ground reaction force data 

collected in the NASA Glenn Research Center enhanced Zero-gravity Locomotion Simulator 

during a study designed to assess harness comfort (Perusek et al. 2007). The following gait 

parameters are illustratively explained within the figure: peak ground reaction force (Peak 

GRF), contact time (Tcontact) and stride time (Tstride). The alternating pulses represent contact by 

the left foot (L) and the right foot (R). Stride frequency (fstride) is defined as the inverse of Tstride

and the stride length is defined as the product of treadmill speed and Tstride. 
NASA/TM—2014-218306 10 

Figure 2.—Representative waveform of the resultant hip joint contact force during one squat. In a 

finite element model of the proximal femur, the force vector would relate to a comparative 

waveform for effective stress or maximum principle strain values in the hip. This figure was 

produced relative to a figure of the in vivo hip joint contact force measured with instrumented 

implants during a knee bend (Bergmann et al. 2001). The percent body weight increased 

uniformly across the cycle by a factor to adjust to approximate values for an adult who is 

moderately loaded while performing a squat. 

Discussion 

The DLS expressions differed in their applications. Expressions I and IV are the result of models fit 

to animal data. Expression III was used to quantify the amount of daily activity to maintain bone density, 

while VI was used to relate bone load from specific activities to the activity of remodeling units in a 

theoretical model of bone remodeling. The mathematical differences that appear in the DLS expressions 

are largely due to the use of the power function m N1/ verses the log(1  N ) , and the role of stress, strain 

or GRF verses the role of strain rate in the form of the series expansion E. Since the quantifiable merits of 

the formulas would have to be based on more data, and in particular, human data, this discussion focuses 

more on the qualitative merits, and tries to answer the following questions: 

(a) As a DLS formula is applied to each of the five exercise scenarios, does the formula demonstrate 

the ability to satisfy assumptions A to E? 

(b) How well does the formula demonstrate the ability to satisfy assumptions A to E? 

 
NASA/TM—2014-218306 11 

Row 1 of Table 2 shows the evaluations of DLS I. Referring to the curve 1/ 4 N in Figure 3; the 

expression shows the diminishing effect of the foot impacts from running (scenario 1) as the number 

increases up to 1200. However, compared to the diminishing effect of log(1  N ) used by DLS IV, 1/ 4 N

does not model saturation as well for large values of N such as those obtained from running. 

Numerically, (1296)1/4 is two times (81)1/4 whereas log (1000) is only one more than log (100). So, 

although Turner (1998) showed how well Carter’s formula could fit to the data of Umemura et al. (1997) 

for values up to N = 100, it may not extend well to a larger range of values. The evaluations of DLS I for 

scenarios 2 to 5 give an identical result, and therefore do not capture the benefits of osteogenic recovery, 

multiple shorter bouts, and rest insertion stated in assumptions C, D, and E. Since the DLS expression II 

is the square root of DLS I, the same analysis applies, it may not work as well for large values of N , and 

fails to satisfy assumptions C, D, and E. 

 

Figure 3.—Comparison of power function to log function. 

o 	 

 + 

   

0 200 400 600 800 1000 1200 1400

0

1

2

3

4

5

6

N

0 5 10 15 20 25 30

0

0.5

1

1.5

2

2.5

N
NASA/TM—2014-218306 12 

In the evaluations of DLS III, note that the application of the algorithm forces complete saturation at 

about 333 impacts on the left or right foot. Although this is the only one of the six expressions that 

prevents any unrealistic contributions to bone response as N becomes increasingly large, it’s unknown if 

saturation after 5 min of running is applicable to humans. The authors of expression III admit this to be a 

limitation in the formula. The evaluation of scenario 3, multiple shorter bouts, correctly applies the 

osteogenic recovery factor in the algorithm but the quantification is effectively the same as scenario 2, a 

single longer bout, thus excluding any benefit from multiple shorter bouts. Since the evaluation of 

scenario 4 is the same as scenario 2, the DLS expression fails to capture any benefit of short rest insertion, 

assumption E. Because of the application of the osteogenic recovery factor to the second bout of exercise 

of scenario 5, contribution to DLS from all loading cycles is only 14. Thus, expression III has the ability 

to distinguish the case when osteogenic potential has not fully recovered following a certain amount of 

saturation, assumption C, and is the only one in column 5 showing that ability. 

The evaluations of DLS IV are the only ones that use log(1  N ) instead of N . As, indicated in 

Figure 1, the running scenario 1 will show a more asymptotic approach as N increases to 1200 and 

beyond with the log function. Also, DLS expression IV is able to capture the benefit of multiple shorter 

bouts as indicated by the scenario 3 evaluation. However, the benefit of short rest insertion, assumption E, 

of scenario 4, is not captured nor is insufficient osteogenic recovery, assumption C, accounted for by the 

scenario 5 evaluation. More importantly, the values E(i) consist of a finite sum of terms of the series 

formed by Equation (11), which can in certain cases have an infinite value. For example, consider the 

trapezoidal strain waveform that Turner (1998) uses to illustrate the formation of the series. The 

waveform is illustrated in Figure 4 and its functional representation is: 







 

!



"

#

 $ $

$ $

$ $  4 2 , 1.5 2

1, 0.5 1.5

2 , 0 0.5

( )

t t

t

t t

f t A (13) 

In the caption of Turner’s figure, a static loading component of about 0.75A is shown followed by 

increasing frequency components with progressively lower amplitudes, but the Fourier series is not 

displayed. The Fourier series expansion of Equation (13) is 

n t

n

n

f t A A

n

)) cos

2 (1 cos(

( )

4 ( ) 0.75

1

2  %

%   %   






 (14)

The infinite series formed from summing the products of strain amplitude and frequency in each term 

becomes 





 

!

"

#

                    

%   0 2

7 1

7

1

2

6 2

6

1

2

5 1

5

1 0

2

3 1

3

1 2 1

2

1

2

1 1

1

4 1

2 2 2 2 2 2 2 E A

 



 

!

"

#

        

%   7

1 2

6

1

5

1 

3

1 2

2

1 1

2

4

2 A 

 






 %&  



 

!

"

#

   

%

& 

1

2 2

1

4

1

2

4 7

1

5

1

3

1

2

4

n n

A A (15)

 
NASA/TM—2014-218306 13 

Figure 4.—Trapezoidal loading waveform.

Note that the series is greater than one fourth of the well-known harmonic series which is divergent, 

so by comparison, the entire series diverges. This suggests revisiting ways to relate bone response to 

multiple higher frequencies. Turner’s cites strong evidence from experimental studies to justify the direct 

relation of a single term, consisting of the product of a peak-to-peak loading magnitude and a frequency, 

on bone formation. This appears true for a sinusoidal loading waveform. As he illustrates, you cannot 

keep loading frequency and strain rate constant while varying strain magnitude (wave height), assuming 

that the duration (wave width) of the strain magnitude remains constant. In Figure 4, if the strain 

magnitude is lowered while keeping the strain rate (slope) constant, the loading frequency will increase. 

So in case of non-sinusoidal loading waveforms, the loading frequency may need to be captured in a 

strain stimulus relation. In those cases, an expression like: 

l   f

where  is the peak strain magnitude and l f is the true loading frequency, may serve as a generalization 

of the simple sinusoidal loading waveform. 

As to row 5 of Table 2, DLS V fails to satisfy assumptions C, D, and E, and also uses the finite sum 

of terms of a potentially divergent series, Equation (11), for the values E(i) . Lastly, the evaluations in 

row 6, scenarios 2 to 5, are identical and indicate that the use of DLS expression VI did not involve the 

consideration of saturation. None of the other effects of rest insertion or osteogenic recovery is able to be 

captured, so it can potentially fail all of the assumptions except A. 

The DLS algorithm presented by Genc et al. (2009) appears to satisfy the most assumptions. 

Although it does not appear explicitly in the expression, it’s the only one that has a temporal dependency. 

However, it does have limitations in applications to human models, as admitted by the authors. The 

numerical time limits for running and walking may not apply to humans. In the specific application 

(Cavanagh et al. 2010), the formula was used to estimate the amount of GRF from exercise prescriptions 

performed on orbit relative to the amount experienced by astronauts in a typical working day on Earth. 

The conclusion from the results was that duration and loading from current ISS exercise prescription do 
NASA/TM—2014-218306 14 

not provide sufficient weight bearing stimulus required to prevent bone loss. It was not intended to predict 

the exact contribution of specific exercise induced stress at a specific skeletal site. However, the 

algorithmic scheme may form the basis for devising a formula suitable for estimating the average amount 

of daily stress/strain at a specific skeletal site due to exercise induced cyclic loading applicable to bone 

remodeling theories. If the GRF in the osteogenic index (OI) measure of exercise effectiveness (Turner 

and Robling 2003) is replaced by effective stress, a DLS can be defined by 






	   

k

j

N j j S

1

ln(1 ) (16)

Use this expression in the algorithm of DLS III, including the limit on duration and the factor accounting 

for osteogenic recovery, and the results of the exercise scenarios become those in Table 3. 

TABLE 3.—EVALUATIONS OF THE EXERCISE SCENARIOS FOR EXPRESSION (15) 

Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5 

r ln(1201)  s ln(21)  s ln(107.4)  s ln(21)  s ln(28) 

This shows a potential of satisfying all the assumptions except E. Stress values in a bone site could come 

from a finite element model with loading boundary conditions obtained from a commercially available or 

open source biomechanics modeling tool. 

Conclusion 

The results of Table 2 indicate that no single DLS expression in Table 1 appears able to satisfy all the 

assumptions A through E (in fact none seem to satisfy assumption E) regarding exercise induced cyclic 

loading. Although the algorithm presented by Genc et al. (2009) appears to satisfy the largest number of 

assumptions (A, B, and C), it does not provide results in units of stress or in terms of strain. Therefore it’s 

unsuitable for direct utilization in the DAP bone adaptation model under development discussed in the 

Introduction. However, its algorithmic scheme may form the basis for devising a robust DLS formula in 

terms of stress or strain like measures as described in combination with an example expression like 

Equation (16). Another option is to combine Genc’s algorithmic scheme with Carter’s expression I in 

Table 2. Ultimately, DLS expressions will be tested in the computational development of the DAP model 

to determine if they adequately represent the osteogenic response of bone to exercise. 

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